Objective function doesn't converge but jumps around

I have been trying to complete an optimisation problem. I am looking at radial heat diffusion on a cylinder. I want to tackle it as an inverse problem in order to obtain values for heat diffusion and the like. I have an objective function Boundary_temperature which spits out the difference between the simulated data(computed by Battery_temp) and the measured data(contained in synth_temp).

The value of the objective function is just the l^2 norm of the calculated data and measured data, so a good value of the parameters should yield a small value. I set up the problem and the value of the objective function jumped about and didn't converge but jumped around, I even managed to obtain a warning of the matrix is singular warning which I don't understand.

I added in a stopping criterion, TolFun, which I set to 10^-3, and I hoped that would make things better, as I know that the value of the objective function did get that small, so I hoped that it would work quite nicely, I was wrong.

The equation in question is:

, ,

The variables I am trying to find is .

[Matt J]

It is hard to tell from the code you've attached what the unknowns are.

[Matthew Hunt]

I've added in some equations to make things clear.